Question

A fair number cube has faces labeled $2,3,4,5,6$ and $7$.

If the number cube is rolled $300$ times, about how many times should the result be a prime number?

• A. $50$
• B. $100$
• C. $150$
• D. $200$

Answer 1

The correct option is D

$200$

Explanation for the correct option:

Step-1: Finding the probability of getting each number after rolling the cube

Given that the cube is fair and its faces are labeled as $2,3,4,5,6,7$.

So, on rolling the cube, each of the six faces can be turned out with the equal probability $p$. Now, the sum of the probabilities will be $1$.

Hence, we must get:

$$\begin{gathered} 6p=1 \\ \Rightarrow p=\frac{1}{6} \end{gathered}$$

Therefore, the probability of getting any one of the six numbers $2,3,4,5,6,7$ is $\frac{1}{6}$.

Step-2: Finding the probability of obtaining a prime number

Out of six numbers $2,3,4,5,6,7$, the primes are : $2,3,5,7$ i.e. there are four prime numbers.

Now, after rolling the cube, the result will be a prime number if any one of these four numbers $2,3,5,7$ turns up each with probability $p=\frac{1}{6}$.

So, the probability of obtaining a prime numbers will be

$$\begin{gathered} =4p \\ =4\times \frac{1}{6} \\ =\frac{2}{3} \end{gathered}$$

Step-3: Calculating the number of times the result is a prime number:

Given that the number cube is rolled $300$ times.

Also, from Step-$2$, we can see that the probability of getting a prime is $\frac{2}{3}$.

Therefore, the number of times of getting a prime number out of $300$ times will be

$$\begin{gathered} =300\times \frac{2}{3} \\ =200 \end{gathered}$$

Hence, option $\left(D\right)$ is the correct answer.